Let us recall Ciarlet's definition of finite elements; see, e.g., [1]}. For nodal finite element methods (continuous or discontinuous), one can define a set of points (so-called nodes) and an associated set of degrees of freedom, corresponding to pointwise evaluation at the nodes. Nodes are chosen such that the degrees of freedom uniquely define functions in \(V_{h}^-\)  and thus define a basis for its dual space. The dual basis of the degrees of freedom are the so-called shape functions. Thus, there is a correspondence between nodes, degrees of freedom and shape functions. We use the following notation: given a node \(\alpha _i\) , we represent its corresponding shape function and degree of freedom by \(\phi _i\)  and \(\sigma _i(\cdot )\) , respectively. The duality between the degrees of freedom and the shape functions implies
\(\sigma _i(\phi _j)=\delta _{ij}\)  with \(\delta _{ij}\)  the Kronecker delta, and the relation between the nodes and the degrees of freedom in turn implies \(\phi _i(\alpha _j)=\delta _{ij}\) . By virtue of the dual relation between degrees of freedom and shape functions, the following correspondence holds for all \(v_h^-\in {}V_h^-\) : \(y_i=\sigma _i(v_h^-)\)  if and only if \(v_h^-=\sum _i y_i \phi _i\) .
